A333896 -id:A333896 - cp's OEIS frontend

A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 5, 3, 1, 1, 1, 5, 8, 10, 3, 1, 1, 1, 6, 14, 26, 16, 4, 1, 1, 1, 7, 20, 61, 60, 29, 4, 1, 1, 1, 8, 30, 128, 243, 184, 45, 5, 1, 1, 1, 9, 40, 254, 800, 1228, 488, 75, 5, 1, 1, 1, 10, 55, 467, 2518, 7252, 6684, 1509, 115, 6, 1

Offset: 0

Andrew Howroyd, Apr 08 2020

T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333737.

Terms may be computed without generating each graph by enumerating the graphs by degree sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A188403. Burnside's lemma as applied in A192517 can be used to extend this method to the unlabeled case.

			Array begins:
==============================================
n\k | 0 1  2   3    4     5      6       7
----+-----------------------------------------
  0 | 1 1  1   1    1     1      1       1 ...
  1 | 1 1  1   1    1     1      1       1 ...
  2 | 1 2  3   4    5     6      7       8 ...
  3 | 1 2  5   8   14    20     30      40 ...
  4 | 1 3 10  26   61   128    254     467 ...
  5 | 1 3 16  60  243   800   2518    6999 ...
  6 | 1 4 29 184 1228  7252  38194  175369 ...
  7 | 1 4 45 488 6684 78063 772243 6254652 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals 0..26)

Rows n=0..4 are A000012, A000012, A000027(n+1), A006918(n+1), A333897.
Columns k=0..5 are A000012, A008619, A000990, A333894, A333895, A333896.
Cf. A188403, A192517, A263293, A333161, A333330, A333737, A334546.

A334547 Number of unlabeled connected loopless multigraphs with n nodes of degree 5 or less.

Original entry on oeis.org

1, 1, 5, 14, 93, 602, 5847, 66289, 937696, 15575285, 301360805, 6663874305, 166516898890, 4654082023569, 144301649717895, 4928171085359360, 184252862642720722, 7501009857590958129, 330928677041195111955, 15754816518916261088582, 806306082299053607173143

Offset: 0

Andrew Howroyd, May 05 2020

nonn

Column k=5 of A334546.

Cf. A303031, A303032, A303033, A333896.

Inverse Euler transform of A333896.

cp's OEIS Frontend

A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.

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A334547 Number of unlabeled connected loopless multigraphs with n nodes of degree 5 or less.

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